arXiv subjects

Hiraku Nozawa

Publications and source records attributed to Hiraku Nozawa.

23 records · Page 2Linked to original sources

arXiv2011

Rigidity of the Álvarez classes of Riemannian foliations with nilpotent structure Lie algebras

We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the Álvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the Álvarez class under deformations shown in arXiv:1009.1098v2, we prove that the Álvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.

math.DG↗

arXiv2011

Continuity of the Alvarez class under deformations

A foliated manifold (M,F) is minimizable if there exists a Riemannian metric g on M such that every leaf of F is a minimal submanifold of (M,g). Alvarez Lopez defined a cohomology class of degree 1 called the Alvarez class of (M,F) whose triviality characterizes the minimizability of (M,F), when M is closed and F is Riemannian. In this paper, we show that the family of the Alvarez classes of a smooth family of Riemannian foliations is continuous with respect to the parameter. Since the Alvarez class has algebraic rigidity under certain topological conditions on (M,F) as the author showed in arXiv:0909.1125, we show that the minimizability of Riemannian foliations is invariant under deformation under the same topological conditions.

math.DG↗

arXiv2010

Rigidity of the Alvarez class

Let $(M,\mathcal{F})$ be a closed manifold with a Riemannian foliation. The Álvarez class is the cohomology class of degree 1 of $M$ whose triviality characterizes the minimizability of $(M,\mathcal{F})$. We show that the integral of the Álvarez class along every closed path in $M$ is the logarism of an algebraic integer if $π_{1}M$ is polycyclic or $\mathcal{F}$ is of polynomial growth.

math.DG↗

arXiv2010

Minimizability of developable Riemannian foliations

Let (M,F) be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino's commuting sheaf of (M,F) vanish if (M,F) is developable and the fundamental group of M is of polynomial growth. By theorems of Álvarez López, our result implies that (M,F) is minimizable under the same conditions. As a corollary, we show that (M,F) is minimizable if F is of codimension 2 and the fundamental group of M is of polynomial growth.

math.DG↗

arXiv2009

Five dimensional K-contact manifolds of rank 2

A K-contact manifold is a smooth manifold M with a contact form whose Reeb flow preserves a Riemannian metric on M. Main examples are Sasakian manifolds. Our results in this paper are four results i), ii), iii) and iv) below obtained by the application of the Morse theory to the contact moment maps on closed 5-dimensional K-contact manifolds of rank 2. 5-dimensional K-contact manifolds of rank 2 have the lowest dimension among K-contact manifolds with nontrivial contact moment maps which are not toric. i) Correspondence between the isomorphism classes of K-contact manifolds and graphs of isotropy data, ii) Classification of K-contact manifolds up to contact blowing up and down, iii) Every closed 5-dimensional K-contact manifold of rank 2 has a Sasakian metric, iv) A sufficient condition for closed 5-dimensional K-contact manifolds of rank 2 to be contact toric.

math.SG↗